OpenTURNS features a generic implementation of the characteristic function for
all its univariate distributions (both continuous and discrete). This default
implementation might be time consuming, especially as the modulus of t gets
high. Only some univariate distributions benefit from dedicated more efficient
implementations.

Where the random vector follows the probability
distribution of interest, and is either the probability
density function of if it is continuous or the
probability distribution function if it is discrete.

OpenTURNS features a generic implementation of the characteristic function for
all its univariate distributions (both continuous and discrete). This default
implementation might be time consuming, especially as the modulus of t gets
high. Only some univariate distributions benefit from dedicated more efficient
implementations.

An iso-probabilistic transformation is a diffeomorphism[1] from
to that maps realizations
of a random vector into realizations
of another random vector while
preserving probabilities. It is hence defined so that it satisfies:

The present implementation of the iso-probabilistic transformation maps
realizations into realizations of a
random vector with spherical distribution[2].
To be more specific:

if the distribution is elliptical, then the transformed distribution is
simply made spherical using the Nataf (linear) transformation[Nataf1962], [Lebrun2009a].

if the distribution has an elliptical Copula, then the transformed
distribution is made spherical using the generalized Nataf
transformation[Lebrun2009b].

otherwise, the transformed distribution is the standard multivariate
Normal distribution and is obtained by means of the Rosenblatt
transformation[Rosenblatt1952], [Lebrun2009c].

The standard representative distribution is defined on a distribution by distribution basis, most of the time by scaling the distribution with bounded support to or by standardizing (ie zero mean, unit variance) the distributions with unbounded support. It is the member of the family for which orthonormal polynomials will be built using generic algorithms of orthonormalization.